USC Predict COVID Project: Predictive Epidemic Model for COVID-19 in Los Angeles County
Abigail Horn, PhD1, Lai Jiang, MS, Wendy Cozen, MD, David Conti, PhD
4/24/2020
Updated with data available as of April 25, 2020
Summary
Aims
The USC Predict COVID project is using an epidemic model to estimate the impact of COVID-19 in Los Angeles County
We are addressing the key questions of:
- When will the peak of the epidemic occur and how will it impact health care capacity?
- What happens to the dynamics of the epidemic when social distancing ends?
- How will the epidemic affect different at-risk groups?
Why our model is unique
- Our epidemic compartmental model uses stochastic differential equations and approximate Bayes calculation techniques for parameter estimation.
- Importantly, the model presents the uncertainty in all estimations and predictions.
- We incorporate prior information for parameter specification.
- We incorporate risk factors (e.g. advanced age, existing health conditions) into the analysis.
- We can modify parameters at different time points, enabling the specification of interventions, e.g. social distancing scenarios
Predictions and Estimation
(1) Critical healthcare variables predicted by the model are the counts of the numbers of individuals over time, including the peak occurrence, for the following:
- The total number of infected cases including both the number detected and observed with testing and the undetected/untested cases
- The total number of individuals hospitalized (including those in the ICU)
- The number of patients in the ICU
- The number of patients on ventilators
- The number of deaths
(2) We estimate a number of key epidemic parameters, including:
- \(R0\), the reproductive number or average number of new infections generated by an infected person in a completely susceptible population
- \(r\), the proportion of illnesses that are detected and reported out of all illnesses
- \(Frac_{R0}\), the reduction in the initial R0 due to social distancing
- \(\alpha\), the probability of hospitalization given illness, i.e. \(Pr(Hospital | Illness)\)
- \(\kappa\), the probability of ICU care necessary given hospitalization, i.e. \(Pr(ICU | Hospital)\)
- \(p_v\), the probability of ventilation given ICU care, i.e. \(Pr(Ventilation | ICU)\)
- \(\delta\), the probability of death given ICU care, i.e. \(Pr(Death | ICU)\)
Projections
Current Projections: Summary of Key Model Estimated Variables
| Median | Upper 50 CI | Lower 50 CI | |
|---|---|---|---|
| Peak Hospitalizations | 6,118.00 | 9,796.00 | 2,554.00 |
| Deaths by August 1, 2020 | 21,863.00 | 31,521.00 | 12,765.50 |
| Detected Illnesses by August 1,2020 | 624,165.00 | 901,032.75 | 364,527.00 |
| Total Illnesses by August 1, 2020 | 5,413,184.00 | 6,716,862.75 | 3,208,621.25 |
| Proportion of Cases Detected (%) | 13.22 | 16.04 | 10.45 |
| CFR Based on Observed Illnesses (%) | 3.87 | 4.77 | 2.78 |
| CFR Based on Total Illnesses (%) | 0.48 | 0.61 | 0.35 |
| R0 - before social distancing | 3.38 | 3.79 | 2.96 |
| % Reduction in Social Contacts (March 15 - ) | 59.82 | 54.26 | 63.59 |
Current Projections: Illnesses, Hospitalizations, ICU Admittances, Ventilation Requirements, Deaths
Dashed line = Maximum possible capacity (i.e., total licensed hospital beds, ICU beds, ventilators) in L.A. County
Model Fits Against Data
Demonstrating model fit against COVID-19 data for Los Angeles, for the following variables:
- Observed illnesses, cumulative
- Hospitalizations, cumulative
- Hospitalization, daily new incidence
- Ventilation, cumulative
- Deaths, cumulative
- Deaths, daily new incidence
COVID-19 data is shown as black dots in the figures below.
Comparing Social Distancing Mitigation Strategies
- Model projections forecast that, if everything continutes as is, LA is not going to see a sharp peak in illnesses in the near future, and rather will see a long, shallow, manageable epidemic curve over the next number of months.
- What can we do to maintain this manageable epidemic curve while sustaining a functioning society?
- Here we evaluate the effect of scenarios relating to social distancing on the ultimate epidemic trajectory
Scenario Comparison
Comparison between current level of social distancing of ~50% beginning March 15, 2020 and counterfactual: if social distancing had never been implemented 
Comparison between current level of social distancing of ~50% beginning March 15, 2020 and easing of social distancing from 50% to 25% on May 1, 2020 
Comparison between current level of social distancing of ~50% beginning March 15, 2020 and easing of social distancing from 50% to 25% on June 1, 2020 
Comparison between current level of social distancing of ~50% beginning March 15, 2020 and easing of social distancing from 50% to 25% on July 1, 2020 
Comparison between current level of social distancing of ~50% beginning March 15, 2020 and removal of social distancing on June 1, 2020 
Comparison between policy of easing of social distancing from 50% to 25% on June 1, 2020 and gradual easing beginning June 1, 2020 of social distancing from 50% to no social distancing 
Projections by Key Risk Groups and Risk Factors
We analyze how population prevalence of known COVID-19 risk factors: advanced age, existence of other health conditions or comorbidities, smoking status, and obesity status, affect COVID-19 illness trajectories in L.A. County and spatial subdivisions.
First, we estimate the conditional probability of COVID illness severity given combinations of risk factors. We categorize the population into a number of risk profiles, representing different combinations of known COVID-19 risk factors: advanced age, existence of other health conditions or comorbidities, smoking status, and obesity status. Using previous COVID-19 studies reporting the marginal risk of severe COVID-19 outcomes given individual risk factors, we develop a statistical model to estimate the probability of COVID illness trajectories for individuals having or not having combinations of risk factors represented by these risk profiles. Specifically, we estimate the probability that individuals within a specific risk group are admitted to hospital given having acquired illness \(Pr(Hospital | Illness, Profile_i)\), are admitted to the ICU given admittance to hospitalized \(Pr(ICU | Hospital, Profile_i)\), and that die given being admitted to the ICU \(Pr(Death | ICU, Profile_i)\). More information is provided below under Methods and Data.
For the analysis below, we have grouped the multiple risk profiles into 5 key risk groups according to similar within-group levels of the probabilities \(Pr(Hospital | Illness, Profile_i)\), \(Pr(ICU | Hospital, Profile_i)\), and \(Pr(Death | ICU, Profile_i)\).
Second, we use these probabilities to estimate the proportion of each risk group that will make up the resulting cohorts of COVID patients admitted to hospital, admitted to ICU, or that die within the L.A. County population, based on the prevalence of each risk group in the population.
Results are also presented for each Service Planning Area (SPA) population within L.A. County. A SPA is a specific geographical region within Los Angeles County used by the Department of Public Health to plan and provide health services. There are 8 SPAs in Los Angeles County.
Risk Profiles, Risk Factors, and Risk Groups
- The following table presents the model-estimated probabilities \(Pr(Hospital | Illness,Profile_i)\), \(Pr(ICU | Hospital,Profile_i)\), and \(Pr(Death | ICU,Profile_i)\) for each risk group (or combination of risk factors), as well as the prevalence of these risk groups/factors in the general L.A. County Population \(Pr(Profile_i)\).
5 Risk Groups by Stage of Disease and SPA
- These figures show the estimated proportion of each risk group that will make up the resulting cohorts of COVID patients admitted to hospital, admitted to ICU, or that die within the L.A. County/SPA population, based on the population prevalence of the risk group in L.A. County/SPAs.
- The analyses are presented for each risk group, as well as stratified to the individual risk factors (age, comorbidities, obesity status, smoking status).
Key Risk Factors by Stage of Disease
Trajectories: Hospitalizations, ICU, Deaths by Key Risk Factors LA County
Trajectories: Broken down by key risk factors for LA County
Parameter estimates
Model estimated parameters and prior information
Here we summarize our estimated parameter values for key epidemic and model quantities:
- \(R0\), the reproductive number or average number of new infections generated by an infected person in a completely susceptible population
- \(r\), the proportion of illnesses that are observed
- \(Frac_{R0}\), the reduction in the initial R0 due to social distancing
- \(\alpha\), the probability of hospitalization given illness, i.e. \(Pr(Hospital | Illness)\)
- \(\kappa\), the probability of ICU care necessary given hospitalization, i.e. \(Pr(ICU | Hospital)\)
- \(p_v\), the probability of ventilation given ICU care, i.e. \(Pr(Ventilation | ICU)\)
- \(\delta\), the probability of death given ICU care, i.e. \(Pr(Death | ICU)\)
Because our model is stochastic and we are using Bayesian techniques for parameter estimation, each posterior parameter estimate is represented by a distribution of likely values.
This table summarizes key statistics of each estimated parameter: the mean and the standard deviation (sd).
| R0 | Prop. cases detected (r) | Frac R0 Mar11 | Pr(Death|ICU) | Pr(Hospital|Illness) | Pr(ICU|Hospital) | Pr(Ventilation|ICU) | Frac R0 Apr23 | |
|---|---|---|---|---|---|---|---|---|
| mean | 3.36 | 0.14 | 0.42 | 0.54 | 0.21 | 0.36 | 0.74 | 0.40 |
| sd | 0.61 | 0.05 | 0.06 | 0.16 | 0.03 | 0.04 | 0.08 | 0.06 |
- Mean = 3.36
- Standard deviation = 0.61
Information informing prior distribution - \(R0\) prior estimate is based on values for \(R0\) estimated from other published studies on COVID-19.
- Mean = 0.14
- Standard deviation = 0.05
- Mean = 0.42
- Standard deviation = 0.06
Information informing this parameter’s prior distribution:
We use mobility data to narrow the specification of the reduction in the average number of new infections due to an infected person (R0) in a completely susceptible population under recent social distancing restrictions.
Effectively, reductions in mobility correspond to a proportional reduction in R0
Reduction in mobility observed in LA County:
Source: Assessing changes in commuting and individual mobility in major metropolitan areas in the United States during the COVID-19 outbreakOur modeled reduction in R0 timeline:
We use previous studies to narrow the specification of the probability of hospitalization given illness, admittance to the intensive care unit (ICU) given being in hospital, ventilation given being in ICU, and death given being in ICU by incorporating risk factors, including age, sex, smoking and other comorbidities. The prevalence of these risk factors in Los Angeles County is also included.
Studies on COVID-19 clinical presentation and trajectories to inform the probability of hospitalization, ICU, and ventilation based on single risk factors: - Guan, Wei-jie, et al. “Clinical characteristics of coronavirus disease 2019 in China.” New England Journal of Medicine (2020). - Petrilli, Christopher M., et al. “Factors associated with hospitalization and critical illness among 4,103 patients with COVID-19 disease in New York City.” medRxiv (2020).
Prevalence data sources: - Los Angeles County Health Survey - UCLA California Health Information Survey
- Mean = 0.21
- Standard deviation = 0.03
- Mean = 0.36
- Standard deviation = 0.04
- Mean = 0.54
- Standard deviation = 0.16
- Mean = 0.74
- Standard deviation = 0.08
Summary
- Predictive epidemic modeling can help to evaluate the public health strategies for limiting the spread of COVID-19. LA County is succeeding in mitigating the epidemic curve due to strong adherence to social distancing (model-estimated reduction in infection rate of >60%).
- Our model suggests that social distancing may need to be continued for a longer period of time to retain these mitigation effects.
- Our model predicts that higher risk groups – characterized not only by advanced age but also combinations of existing health conditions – will make up the majority of those afflicted with severe disease. This model estimates how disparities in prevalences of risk factors across neighborhoods within LA will impact severe disease outcomes.
- Model estimates can and will change as new data comes in.
Methods and data
Data
Illness case data
- We use current numbers of infected (observed), hospitalized, ICU, ventilated, and deaths, as well as the capacity (total number of resouces available) at hospitals, ICUs, and ventilators in L.A. County from the Los Angeles County Department of Public Health, updated daily by Faith Washburn (shared privately)
- The data used in this version of the report (April 24th, 2020) includes counts up through April 22th, 2020
Model overview
Stochastic differential equation model:
- Deterministic model: for given values of parameters, dynamics across compartments will be fixed.
- Stochastic model: Appropriate probability distributions are used model the transfer of individuals across compartments.
What a stochastic model allows:
- Looking back: Provides a better framework for parameter estimation, based on observed data
- Looking forward: Enables forecasts with confidence bounds that account for variability in parameters
Approximate Bayesian Computation (ABC) for parameter estimation:
- Allows us to incorporate the uncertainty in all the parameters in fitting the model to data and estimating parameters
- Allows us to include all prior information and/or assumptions about the distribution (the range of values) for each parameter
- Allows us to prioritize data input that is more reliable in fitting the model to data (e.g., not including more unreliable early illness count data)
Flow diagram
Compartmental model flow diagram
System of Equations
\[ \begin{align*} dS/dt &= -\beta S(I+A)\\ dE/dt &= \beta S(I+A) - \tfrac{1}{d_{EI}}E\\ dA/dt &= \tfrac{1-r}{d_{EI}}E - \tfrac{1}{d_{IR}}A\\ dI/dt &= \tfrac{r}{d_{EI}}E - (\tfrac{\alpha}{d_{IH}}\tfrac{1-\alpha}{d_{IR}})I\\ dH/dt &= \alpha (\tfrac{\alpha}{d_{IH}}\tfrac{1-\alpha}{d_{IR}})I - (\tfrac{\kappa}{d_{HQ}}\tfrac{1-\kappa}{d_{HR}})H \\ dQ/dt &= \kappa (\tfrac{\kappa}{d_{HQ}}\tfrac{1-\kappa}{d_{HR}})H - (\tfrac{\delta}{d_{QD}}\tfrac{1-\delta}{d_{QR}})Q \\ dV/dt &= p_V Q\\ dD/dt &= \delta (\tfrac{\delta}{d_{QD}}\tfrac{1-\delta}{d_{QR}})Q\\ dR/dt &= (1-\alpha) (\tfrac{\alpha}{d_{IH}}\tfrac{1-\alpha}{d_{IR}})I + (1-\kappa) (\tfrac{\kappa}{d_{HQ}}\tfrac{1-\kappa}{d_{HR}})H + (1-\delta)(\tfrac{\delta}{d_{QD}}\tfrac{1-\delta}{d_{QR}})Q + \tfrac{1}{d_{IR}}A \ \end{align*} \]
\[ R0 = \beta ({\frac{r}{\tfrac{\alpha}{d_{IH}}+\tfrac{1-\alpha}{d_{IR}}}+ (1-r){d_{IR}}}) \\ N=S+E+A+I+H+Q+D+R \]
Model parameters
| Parameter | Description | Value |
|---|---|---|
| \(R0\) | Basic reproductive number | Estimated |
| \(\beta\) | transmission rate | Analytically derived from model and R0 |
| \(d_{EI}\) | days between exposure and infectivity (incubation period) | 5 days |
| \(d_{IH}\) | days between symptom onset and hospitalization (if required) | 10 days |
| \(d_{IR}\) | days between symptom onset and recovery (if not hospitalized) | 7 days |
| \(d_{HQ}\) | days between hospitalization and ICU (if required) | 1 days |
| \(d_{QR}\) | days between hospitalization and recovery (if ICU not required) | 12 days |
| \(d_{QD}\) | days between ICU and fatality | 8 days |
| \(d_{QR}\) | days between ICU and recovery | 7 days |
| \(\alpha\) | probability infected (I) requires hospitalization (vs. recovers) | Estimated |
| \(\kappa\) | probability hospitalized (H) requires ICU (vs. recovers) | Estimated |
| \(\delta\) | probability ICU (Q) patient dies | Estimated |
| \(p_V\) | probability ventilation (V) required given ICU | Estimated |
| \(N\) | Total population size | |
| \(S\) | Susceptible population | |
| \(E\) | Exposed not yet infectious | |
| \(A\) | Infected, unobserved | |
| \(I\) | Infected, observed | |
| \(H\) | In Hospital | |
| \(Q\) | In ICU | |
| \(V\) | On ventilator | |
| \(D\) | Dead | |
| \(R\) | Recovered/removed |
Model parameters - fixed, taken from literature: - Transition times between compartments - Sources provided at this link
Model Parameters — estimated by our model - \(R0\), the reproductive number or average number of new infections generated by an infected person in a completely susceptible population - \(r\), the proportion of illnesses that are observed - \(Frac_{R0}\), the reduction in the initial R0 due to social distancing - \(\alpha\), the probability of hospitalization given illness, i.e. \(Pr(Hospital | Illness)\) - \(\kappa\), the probability of ICU care necessary given hospitalization, i.e. \(Pr(ICU | Hospital)\) - \(p_v\), the probability of ventilation given ICU care, i.e. \(Pr(Ventilation | ICU)\) - \(\delta\), the probability of death given ICU care, i.e. \(Pr(Death | ICU)\)
Projections by Key Risk Groups and Risk Factors
We analyze how population prevalence of known COVID-19 risk factors: advanced age, existence of other health conditions or comorbidities, smoking status, and obesity status, affect COVID-19 illness trajectories in L.A. County and spatial subdivisions.
(1) Estimating the conditional probability of COVID illness severity given combinations of risk factors
- Using studies reporting the marginal risk of severe COVID-19 outcomes given individual risk factors, we develop a statistical model to estimate the probability of COVID illness trajectories for individuals with combinations of risk factors.
- Specifically, we estimate the probability that individuals having (or not) combinations of risk factors are admitted to hospital given having acquired illness \(Pr(Hospital | Illness)\), are admitted to the ICU given admittance to hospitalized \(Pr(ICU | Hospital)\), and that die given being admitted to the ICU \(Pr(Death | ICU)\).
- Our methodology for joint risk factor estimation relies on a model, called JAM, developed for use in genome-wide analyses to identify the conditional relative risk (RR) of phenotypic occurrence given joint combinations of genes from two pieces of information: (i) the marginal RR between single genes and phenotype and (ii) the correlation structure between the genes.
- We apply the JAM model to estimate the conditional RR of COVID-19 illness severity (hospitalization, ICU, and death) given joint combinations of risk factors. For information informing (i) we obtain the marginal RR between individual risk factors and COVID-19 illness severity from published COVID-19 studies (sources below, peer-reviewed where available). For (2), we obtain the correlation structure between the risk factors using data from The National Health and Nutrition Examination Survey (NHANES). NHANES is a survey research program conducted by the National Center for Health Statistics (NCHS) to assess the health and nutritional status of adults and children in the United States, and to track changes over time. We use the NHANES cohort of 2018-2019.
- After obtaining the conditional RR of COVID-19 illness severity (hospitalization, ICU, and death) given joint combinations of risk factors, we convert these to the probabilities \(Pr(Hospital | Illness, Profile_i)\), \(Pr(ICU | Hospital,Profile_i)\), and \(Pr(Death | ICU,Profile_i)\).
- The analysis we present here has taken the combinations of risk factors and grouped these into 5 key risk groups according to similar within-group levels of the probabilities \(Pr(Hospital | Illness, Profile_i)\), \(Pr(ICU | Hospital, Profile_i)\), and \(Pr(Death | ICU, Profile_i)\).
- To produce an estimate of the overall \(Pr(Hospital | Illness)\) across all risk groups for L.A. County and subpopulations, we take the weighted average of the probability for each risk group and the prevalence of the risk group in all illnesses, i.e.
\[ \begin{align*} Pr(Hospital | Illness) = \sum_i Pr(Group_i | Illness)Pr(Hospital|Group_i,Illness) \end{align*} \] We assume that the prevalence of the risk group in the ill population, \(Pr(Group_i|Illness)\), is equal to the prevalence of the group in the general population of L.A. County, i.e. \(Pr(Group_i)\). We again borrow the correlation structure between risk factors derived from the NHANES cohort to estimate the population prevalence \(Pr(Group_i)\) from available data on the prevalence of individual risk factors.
The same approach is applied to estimate \(Pr(ICU | Hospital)\) and \(Pr(Death|ICU)\): \[ \begin{align*} Pr(ICU | Hosptial) = \sum_i Pr(Group_i | Hospital)Pr(ICU|Group_i,Hospital)\\ Pr(Death | ICU) = \sum_i Pr(Group_i | ICU)Pr(Death|Group_i,ICU) \end{align*} \]
(2) Estimating the proportion of each risk group that will make up the cohorts of COVID patients admitted to hospital, admitted to ICU, or that die in L.A. County and SPAs
- We estimate the proportion of each risk group that will make up the cohorts of COVID patients admitted to hospital (\(Pr(Group_i|Hospital\))), admitted to ICU (\(Pr(Group_i|ICU\))), or death (\(Pr(Group_i|Death\))) in L.A. County population and each SPA, as the relative share of each group and disease status, i.e. \[ \begin{align*} Pr(Group_i|Hospital) = \frac{Pr(Group_i|Illness)Pr(Hospital|Group_i,Illness)}{Pr(Hospital|Illness)}\\ Pr(Group_i|ICU) = \frac{Pr(Group_i|Hospital)Pr(ICU|Group_i,Hospital)}{Pr(ICU|Hospital)}\\ Pr(Group_i|Death) = \frac{Pr(Group_i|ICU)Pr(Death|Group_i,ICU)}{Pr(Death|ICU)}\\ \end{align*} \] We multiply the group prevalences above by the trajectory median values for number in hospital, ICU, and death from the epidemic model to arrive at the trajectory breakdowns presented above.
Data sources
- The comorbidities included in the “any comorbidity” category are: hypertension, diabetes, cardiovascular disease, cerebrovascular disease / stroke, cancer, COPD, and athsma. It is important to note that each of these conditions has its own risk of disease outcome; future models will disaggregate risk estimates by specific risk conditions when sufficient data becomes available to produce these estimations. Studies on COVID-19 clinical presentation and trajectories to inform the probability of hospitalization, ICU, and ventilation based on single risk factors:
- Guan, Wei-jie, et al. “Clinical characteristics of coronavirus disease 2019 in China.” New England Journal of Medicine (2020).
- Petrilli, Christopher M., et al. “Factors associated with hospitalization and critical illness among 4,103 patients with COVID-19 disease in New York City.” medRxiv (2020).
- Summary table of the modified Relative Risks (RR) coming from these studies, which are used by our statistical model estimates
Limitations
- This model does not account for differences in contact patterns within and across key groups, such as workplaces, schools, and communities; or of group-specific social distancing scenarios.
- This model does not account for the role of interventions beyond social distancing, such as contact tracing and surveillance testing, which are critical to identifying and limiting the spread of the virus.
- This model focuses only on the direct effects of stay-at-home orders on COVID-19 infection transmission and hospitalization. It does not account for the non-COVID-19 related public health impact of stay-at-home orders.
- This model does not account for unanticipated behavioral responses to the effects of interventions or mitigation strategies
- New information about the epidemiological characteristics of COVID-19 is continuously arising, and incorporating the latest information into our models is possible and will be key to maintaining its relevance for decision insights.
Coming soon
- Projections by SPA
- Projections by race/ethnicity
Acknowledgements
- We would like to acknowledge Claire Jacquillat and Adam Taylor for their contributions related to data wrangling, cleaning, interpretation, and visualization.
Appendix
Specification of stochastic model
Specification of stochastic model
## ```r
##
## # TRANSITION EQUATIONS
##
## ## Core equations for transitions between compartments:
## update(S) <- S - n_SE
## update(E) <- E + n_SE - n_Eout
## update(I) <- I + n_EoutI - n_Iout
## update(A) <- A + n_EoutA - n_AR
## update(H) <- H + n_IoutH - n_Hout
## update(Q) <- Q + n_HoutQ - n_Qout
## update(D) <- D + n_QoutD
## update(R) <- R + n_IoutR + n_HoutR + n_QoutR + n_AR
##
## ## Htot = H + Q
## update(Htot) <- H + Q + n_IoutH - n_HoutR - n_Qout # Htot represents all in Hospital: Non-ICU + ICU
##
## ## Ventilators (tracking as frac of Q, do not go to other compartments)
## update(V) <- p_QV*Q #V + n_QV - n_Vout
##
## ## Tracking cumulative numbers in compartments:
## update(Idetectcum) <- Idetectcum + n_EoutI
## update(Itotcum) <- Itotcum + n_Eout
## update(Htotcum) <- Htotcum + n_IoutH #Htotcum represents cumulative of all in Hospital: Non-ICU + ICU
## update(Qcum) <- Qcum + n_HoutQ
## update(Vcum) <- p_QV*Qcum #Vcum + n_QV
##
## ## New daily numbers
## output(I_detect_new) <- n_EoutI
## output(I_tot_new) <- n_Eout
## output(H_new) <- n_IoutH
## output(Q_new) <- n_HoutQ
## output(D_new) <- n_QoutD
## #output(d_EI_rand) <- d_EI
##
## ####################################################################################
##
## # PROBABILITIES
##
## ## Individual probabilities of transition:
## p_SE <- 1 - exp(-(Beta * (I+A)) / N) # S to E
## p_Eout <- 1 - exp(-1/d_EI) # E to I
## p_Iout <- 1 - exp(-((Alpha/d_IH) + ((1-Alpha)/d_IR))) #exp(-((1/d_IH) + (1/d_IR))) # I to H and R
## p_Hout <- 1 - exp(-((Kappa/d_HQ) + ((1-Kappa)/d_HR))) #exp(-((1/d_HQ) + (1/d_HR))) # H to Q and R
## p_Qout <- 1 - exp(-((Delta/d_QD) + ((1-Delta)/d_QR))) #exp(-((1/d_QD) + (1/d_QR))) # Q to D and R
## p_AR <- 1 - exp(-1/d_IR)
## #p_Vout <- 1 - exp(-1/d_V) # Leaving V
##
##
##
## # RANDOM DRAWS FOR NUMBERS CHANGING BETWEEN COMPARTMENTS
## ## Draws from binomial and multinomial distributions for numbers changing between compartments:
##
## ### S to E
## n_SE <- rbinom(S, p_SE)
##
## ### E to I and A
## n_Eout <- rbinom(E, p_Eout)
## n_EoutIA[] <- rmultinom(n_Eout, p_EoutIA)
## p_EoutIA[1] <- r
## p_EoutIA[2] <- 1-r
## dim(p_EoutIA) <- 2
## dim(n_EoutIA) <- 2
## n_EoutI <- n_EoutIA[1]
## n_EoutA <- n_EoutIA[2]
##
## ### A to R
## n_AR <- rbinom(A, p_AR)
##
## ### I to H and R
## n_Iout <- rbinom(I, p_Iout) # Total no. leaving I
## n_IoutHR[] <- rmultinom(n_Iout, p_IoutHR) # Divide total no. leaving I into I->H and I->R
## p_IoutHR[1] <- Alpha #(Alpha/d_IH)/((Alpha/d_IH) + ((1-Alpha)/d_IR)) # Goes to H and R with relative rates
## p_IoutHR[2] <- 1-Alpha #((1-Alpha)/d_IR)/((Alpha/d_IH) + ((1-Alpha)/d_IR)) # 1-p_IoutHR[1]
## dim(p_IoutHR) <- 2
## dim(n_IoutHR) <- 2
## n_IoutH <- n_IoutHR[1] # Total no. I->H
## n_IoutR <- n_IoutHR[2] # Total no. I->R
##
## ### H to Q and R
## n_Hout <- rbinom(H, p_Hout)
## n_HoutQR[] <- rmultinom(n_Hout, p_HoutQR)
## p_HoutQR[1] <- Kappa #(Kappa/d_HQ)/((Kappa/d_HQ) + ((1-Kappa)/d_HR))
## p_HoutQR[2] <- 1-Kappa #((1-Kappa)/d_HR)/((Kappa/d_HQ) + ((1-Kappa)/d_HR))
## dim(p_HoutQR) <- 2
## dim(n_HoutQR) <- 2
## n_HoutQ <- n_HoutQR[1]
## n_HoutR <- n_HoutQR[2]
##
## ### Q to D and R
## n_Qout <- rbinom(Q, p_Qout)
## n_QoutDR[] <- rmultinom(n_Qout, p_QoutDR)
## p_QoutDR[1] <- Delta #(Delta/d_QD)/((Delta/d_QD) + ((1-Delta)/d_QR))
## p_QoutDR[2] <- 1-Delta #((1-Delta)/d_QR)/((Delta/d_QD) + ((1-Delta)/d_QR))
## dim(p_QoutDR) <- 2
## dim(n_QoutDR) <- 2
## n_QoutD <- n_QoutDR[1]
## n_QoutR <- n_QoutDR[2]
##
## ### Q to V and Vout
## #n_QV <- rbinom(Q, p_QV)
## #n_Vout <- rbinom(V, p_Vout)
##
## ######################################################################
##
## # TOTAL POPULATION SIZE
## N <- S + E + I + A + H + Q + D + R
##
## ######################################################################
##
## # INITIAL STATES
## ## Core compartments
## initial(S) <- S_ini
## initial(E) <- E_ini
## initial(I) <- 0
## initial(A) <- 0
## initial(H) <- 0
## initial(Q) <- 0
## initial(D) <- 0
## initial(R) <- 0
## initial(V) <- 0
## initial(Htot) <- 0
##
## ## Cumulative counts
## initial(Idetectcum) <- 0
## initial(Itotcum) <- 0
## initial(Htotcum) <- 0
## initial(Qcum) <- 0
## initial(Vcum) <- 0
##
## ######################################################################
##
## # USER DEFINED PARAMETERS
## ## Default in parentheses:
##
## ### Initial conditions
## S_ini <- user(1e7) # susceptibles
## E_ini <- user(10) # infected
##
## ### Parameters - random
## #d_EI <- runif(3, 8)
##
## ### Parameters - fixed
## d_EI <- user(5.2) #days between exposure and infectivity (incubation period)
## d_IH <- user(10) #days between illness onset and hospitalization
## d_IR <- user(7) #days between illness onset and recovery (hospitalization not required)
## d_HQ <- user(1) #days between hospitalization start and ICU
## d_HR <- user(12) #days in hospital (ICU not required)
## d_QD <- user(8) #days in ICU before death (given death)
## d_QR <- user(7) #days in ICU before recovery (given recovery)
## #d_V <- user(3) #days on ventilator (within ICU)
##
## ### Parameters - weighted average risk probabilities: input from JAM + population prevalence
## Alpha <- user(0.14) #probability infected (I) requires hospitalization (vs. recovers)
## Kappa <- user(0.23) #probability hospitalized (H) requires ICU (vs. recovers)
## Delta <- user(0.06) #probability ICU (Q) patient dies
## p_QV <- user(0.667) #probability in ICU and requires ventilation
## r <- user(0.25)
##
## ### Other variables
## #R0 <- user(2.2) #Current estimates from other models
##
## ### Parameters - calculated from inputs
## #Br <- R0 * ( 1 / ( (r/ ((Alpha/d_IH) + ((1-Alpha)/d_IR))) + (1-r)*d_IR ))
##
##
##
## #########################################
## ### TIME VARYING BETA (INTERPOLATION) ###
## #########################################
##
## Beta <- interpolate(Beta_t, Beta_y,"linear")
##
## Beta_t[] <- user()# R0 * ((Alpha/d_IH)+((1-Alpha)/d_IR))
## Beta_y[] <- user()
## dim(Beta_t) <- user()
## dim(Beta_y) <- user()
##
##
##
## ```